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In this paper we develop a new, computationally friendly approach to Lie coalgebras through graph coalgebras, and we apply this approach to Harrison homology. There are two standard to presentations of a Lie algebra through “simpler” algebras. One is as a quotient of a non-associative binary algebra by Jacobi and anti-commutativity identities. Another presentation is as as embedded as Hopf algebra primitives in an associative universal enveloping algebra. The standard presentation of Lie coalgebras in the literature is dual to the second of these – as a quotient of the associative coenveloping coalgebra, namely the Hopf algebra indecomposables [12, 16]. We describe an approach to Lie coalgebras indiginous to the realm of coalgebras, dual to neither of these. We define a new kind of coalgebra structure, namely anti-commutative graph coalgebras, and we show that Lie coalgebras are quotients of these graph coalgebras. Our approach through graph coalgebras gives a presentation for Lie coalgebras which works better than the classical presentation in two respects. First, cofree graph coalgebras come with a simple and easily computable pairing with free binary nonassociative algebras which passes to Lie coalgebras and algebras, making duality not just a theoretical statement but an explicitly computable tool. Secondly, the quotient used to create Lie coalgebras from graph coalgebras is a locally defined relation. The quotient creating Lie coalgebras from associative coalgebras is the shuffle relation, which causes global changes to an expression. As a result, proofs in the realm of Lie coalgebras are often simpler to give through graph coalgebras than through associative coalgebras, and for some important statements we have only found proofs in the graph coalgebra setting. For applications, we investigate the word problem for Lie coalgebras, we find a canonical additive basis for cofree Lie coalgebras (which is surprising since standard bases of free Lie algebras involve choices), and we revisit Harrison homology. The category of graph coalgebras, and the graph cooperad on which it is based, may also be of intrinsic interest. The graph cooperad is not binary, but could play a similar role in some natural category of cooperads as is played by the tree operad for binary operads.

Lie coalgebras and rational homotopy theory, I: graph coalgebras